Abstract. If Y is a diagram of spectra indexed by an arbitrary
poset C together with a specified sub-poset D, we define the total
cofibre Γ(Y ) of Y as cofibre(hocolimD (Y ) - hocolimC (Y )). We
- hom(Z, Γ̂(Y )) to
construct a comparison map Γ̂Y : holimC Y
a mapping spectrum of a fibrant replacement of Γ(Y ) where Z
is a simplicial set obtained from C and D, and characterise those
poset pairs D ⊂ C for which Γ̂Y is a stable equivalence. The
characterisation is given in terms of stable cohomotopy of spaces
related to Z. For example, if C is a finite polytopal complex with
|C| ∼
= B m a ball with boundary sphere |D|, then |Z| ∼
=P L S m ,
and Γ̂(Y ) and holimC (Y ) agree up to m-fold looping and up to
stable equivalence. As an application of the general result we give
a spectral sequence for π∗ (Γ(Y )) with E2 -term involving higher
derived inverse limits of π∗ (Y ), generalising earlier constructions
for space-valued diagrams indexed by the face lattice of a polytope.